Reduced Elliptic Curve Group - E127(-1,3)

This section provides an example of a reduced Elliptic Curve group E127(-1,3). An example of addition operation is also provided.

Let's take a look at another reduced elliptic curve group, E127(-1,3), as discussed in "Elliptic Curve Cryptography: finite fields and discrete logarithms" by Andrea Corbellini at andrea.corbellini.name/2015/05/23/elliptic-curve-cryptography-finite-fields-and-discrete-logarithms/.

Here is the reduced elliptic curve group using modular arithmetic of prime number 127, E127(-1,3):

```   y2 = x3 - x + 3 (mod 127)
```

The above diagram provides all points in this group. It also illustrates an example of the reduced addition operation:

```Given two points on the curve:
P = (16,20)
Q = (41,120)

Draw a straight line passing through A and B,
And wrap the line around when it reaches the boundary of the region.
It will reach another point -R on the curve.
Take the symmetric point R of -R:
R = A + B
R = (86,81)
```

Let's verify R = P+Q using algebraic equations given by the reduced elliptic curve group definition:

```For any two given points on the curve:
P = (xP, yP) = (16,20)
Q = (xQ, yQ) = (41,120)

R = P + Q is a third point on the curve:
R = (xR, yR)

Where:
xR = m2 - xP - xQ (mod p)     (11)
yR = m(xP - xR) - yP (mod p)  (12)
m(xP - xQ) = yP - yQ (mod p)  (18)

Calculation:
m*(16-41) = 20-120 (mod 127)
m*(-25) = -100 (mod 127)
102*m = 27 (mod 127)
m = 27 * 1/102 (mod 127)
m = 27*66 (mod 127)
m = 1782 (mod 127)
m = 4

xR = 4*4 - 16 - 41 = -41 (mod 127)
xR = 86

yR = 4*(16 - 113) - 20 = -300 (mod 127)
yR = 81

C = (86,81)
```

The result from algebraic equations matches the geometrical result!

Last update: 2019.